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Token on a grid

Brilli the ant randomly placed a token into a square on a \(2 \times 100\) chessboard according to a probability distribution \(P.\) The token is then moved uniformly at random to one of the horizontally, vertically, or diagonally adjacent squares. The probability that the token is in a particular position after it has been moved also satisfies the distribution \(P.\) Let \(q\) be the probability that the token is placed into one of the columns of \(C = \{5,6, \ldots 44\}\) and after being moved is still in one of those columns. The value of \(q\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b?\)

Details and assumptions

\(C\) is the set of integers from 5 to 44 (inclusive).

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